Tor-vanishing conjecture for modules of Loewy length at most two

Let $R$ be a commutative Artinian local ring, with maximal ideal $\mathfrak m$, and let $M,N$ be nonzero $R$-modules such that

$$\mathfrak m^2M=\mathfrak m^2N=0.$$

Tor-vanishing conjecture. If

$$\operatorname{Tor}_i^R(M,N)=0$$

for all $i>0$, then $\mathfrak m^3=0$. The conjecture proposes that total positive Tor-vanishing for two nonzero modules annihilated by the square of the maximal ideal forces the ring itself to have Loewy length at most three. The source notes that it holds for a large class of rings, including standard graded rings, complete intersections of codimension greater than $2$, Koszul rings, and Golod rings; the general case remains open there.